`rr_1 cot (A//2)=`

Options

A. `Delta`
B. `Delta^2`
C. `Delta/2`
D. `2 Delta`

Answer

To solve the expression \( rr_1 \cot \left( \frac{A}{2} \right) \), we need to understand the definitions of \( R \), \( R_1 \), and \( \delta \) in the context of triangles.

### Step-by-Step Solution:

1. **Understand the Notations**:
- Let \( \delta \) be the area of the triangle.
- Let \( R \) be the circumradius of the triangle, given by the formula:
\[ R = \frac{\delta}{s} \]
where \( s \) is the semi-perimeter of the triangle.
- Let \( R_1 \) be the inradius of the triangle, given by:
\[ R_1 = \frac{A}{2} \cdot s \]
where \( A \) is the length of side opposite to angle \( A \).

2. **Substituting the Values**:
- We substitute \( R \) and \( R_1 \) into the expression \( rr_1 \cot \left( \frac{A}{2} \right) \):
\[ rr_1 = R \cdot R_1 = \left( \frac{\delta}{s} \right) \cdot \left( s \cdot \frac{A}{2} \right) \]

3. **Simplifying the Expression**:
- The \( s \) in the numerator and denominator cancels out:
\[ rr_1 = \delta \cdot \frac{A}{2} \]

4. **Using the Cotangent Identity**:
- Now we need to multiply by \( \cot \left( \frac{A}{2} \right) \):
\[ rr_1 \cot \left( \frac{A}{2} \right) = \delta \cdot \frac{A}{2} \cdot \cot \left( \frac{A}{2} \right) \]

5. **Final Expression**:
- Since \( \cot \left( \frac{A}{2} \right) = \frac{\cos \left( \frac{A}{2} \right)}{\sin \left( \frac{A}{2} \right)} \), we can express the final result as:
\[ rr_1 \cot \left( \frac{A}{2} \right) = \delta \cdot \frac{A}{2} \cdot \cot \left( \frac{A}{2} \right) \]

### Conclusion:
Thus, the value of \( rr_1 \cot \left( \frac{A}{2} \right) \) simplifies to:
\[ \delta \cdot \frac{A}{2} \cdot \cot \left( \frac{A}{2} \right) \]